How to construct nonlinear ODE systems with given condition?

I have a general question about how to construct nonlinear ODE systems with given condition such as # of critical points with certain characteristics of the phase portrait of each critical point.

I have no problem solving any type of nonlinear ODE system. But to do the reverse order, I have hard time to find an appropriate nonlinear general system to start with.

If I find the right one, then I can proceed to find the critical points with all unknown constants. Then linearize them individually through Jacobian matrix. Use the given characteristics of the critical points based on the eigenvalues from Jacobian matrix to find the parameters of the unknown constants, then randomly choose the constant within the constraints to find a nonlinear system.